Timeline for Is Schauder's conjecture resolved?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 14, 2022 at 15:13 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
|
| Oct 14, 2022 at 13:09 | answer | added | Doug Liu | timeline score: 4 | |
| Nov 2, 2016 at 15:08 | answer | added | jaco | timeline score: 6 | |
| Jul 25, 2016 at 23:14 | comment | added | Gerry Myerson | [continued from previous comment] These results are mainly concerned with generalized vector equilibrium problems.'' | |
| Jul 25, 2016 at 23:13 | comment | added | Gerry Myerson | Sehie Park also has a more recent publication, Remarks on fixed point and generalized vector equilibrium problems, Nonlinear Anal. Forum 20 (2015), 33–41; the summary, as found at MR3444052, goes, "In the first half of this paper, we introduce the contents of some of our previous papers on fixed point problems related to the Schauder conjecture. Some of them contain incorrect statements. The second half devotes to improve or correct the results in certain papers of other authors based on one of our incorrect statements related to a fixed point conjecture. [continued] | |
| Jul 25, 2016 at 22:57 | history | edited | Myshkin |
+ top level tag (gn.)
|
|
| Mar 23, 2015 at 14:58 | comment | added | Johannes Hahn | This could have all been found out (and resolved?) more quickly if the paper wasn't a.) behind a paywall no german university that I've tried had access to at the time and b.) written in french | |
| Mar 23, 2015 at 9:01 | answer | added | Jochen Wengenroth | timeline score: 12 | |
| May 19, 2014 at 15:09 | comment | added | 57319 | In 2005 paper, Park claimed this. But at that time, the gap was not discovered. It is discovered in 2005. And 3 years later, the same author, in "Compact Browder maps and equilibria of abstract economies, Journal of Applied Mathematics and Computing, 26, 555-564 (2008)" wrote: "In 2001, Cauty claimed to resolve the conjecture affirmatively, but later, it was known that his proof had a gap." | |
| May 19, 2014 at 7:12 | comment | added | Mohammad Golshani | I have taken the following from the review of the following paper "Schuader's conjecture on convex metric spaces" written in 2010 : One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is: Schauder's Conjecture. Let E be a compact convex subset in a topological vector space. Then any continuous mapping f:E→E has a fixed point. So it seems that the problem in its general form is still open. | |
| May 18, 2014 at 13:52 | comment | added | Mohammad Golshani | The present author uses this result to obtain some more or less obvious generalizations to the case of set-valued mappings. Here, he reports on rumors that both Cauty's and Dobrowolski's proofs might contain gaps (without giving any evidence for this claim) and he reconsiders his results under the assumption that the rumors should prove correct. | |
| May 18, 2014 at 13:52 | comment | added | Mohammad Golshani | >Park, Sehie; Remarks on recent results in analytical fixed point theory. Nonlinear analysis and convex analysis, 517–525, Yokohama Publ., Yokohama, 2007 R. Cauty proved the Schauder fixed point theorem in topological vector spaces without assuming local convexity (see also T. Dobrowolski, [Revisiting Cauty's proof of the Schauder conjecture] for an expanded version which is more easily accessible). | |
| May 18, 2014 at 8:02 | answer | added | Mohammad Golshani | timeline score: 16 | |
| May 12, 2014 at 6:33 | comment | added | Nate Eldredge | @PerAlexandersson: Wikipedia says 1930, which is when Schauder proved the theorem for Banach spaces (the well-known Schauder fixed point theorem). Tychonoff extended it to locally convex spaces soon after. Note that Wikipedia calls the above conjecture a theorem, apparently accepting Cauty's argument as a proof. | |
| May 11, 2014 at 21:18 | comment | added | Per Alexandersson | Wow, how old is this? This conjecture should have more attention; so simple to state, and clearly has a lot of impact... | |
| May 11, 2014 at 16:31 | history | asked | 57319 | CC BY-SA 3.0 |